Static Pressure Distribution
6.3 MODELLING AND RESULTS
The operating pressure is directly linked to the choice of using the fluid as an "ideal-gas", as this law incorporates the equation:
Pop + P
p= RT
(6.3.1.2.1)
wherePopis the operating pressure,Pis the local relative (gauge) pressure,Ris the universal gas constant of 287 Pa.m3/kg.K, and T is the temperature. The operating pressure must be set as defined by the Mach number observed in the system. The FLUENT user manual suggests the use of the operating pressure set to 0 Pa when M>0.1. This is suited to the operating conditions inside the plenum, and thus the operating pressure was set to 0 Pa. Thus it must be noted that all pressure values entered into and displayed by the program are relative to this operating pressure.
Suction Side
Pressure Side
Pressure Outlet
TB - Test blade Pressure Inlet
Figure 6-2: GAMBIT cascade geometry
The pressure inlet and outlet settings in the program were set using the values attained in the static pressure testing, described in Section 5.2 of Chapter 5. The inlet conditions were obtained from the pressure and temperature probes, and the settings entered into FLUENT are shown in Table 6-1, and applied to the inlet section as shown in Figure 6-2.
"Intensity and Hydraulic Diameter", where the turbulence intensity was measured to be 3% for a free inlet by Snedden (1995) and the hydraulic diameter being the characteristic width of the inlet.
Table 6-1: Inlet Conditions
SETTING VALUE
Total Gauge Pressure 40000 Pa
Supersonic/Initial Gauge Pressure 35000 Pa
Total Temperature 373 K
Direction Specification Method Normal to Boundary
Turbulence Specification Method Intensity and Hydraulic Diameter
Turbulence Intensity 3%
Hydraulic Diameter 118.72 mm
Similarly, the outlet conditions were set by using experimental data and entering the values into the program, applied to the outlet section as seen in Figure 6-2. These settings are shown in Table 6-2.
Table 6-2: Outlet Conditions
SETTING VALUE
Total Gauge Pressure 27000 Pa
Backflow Total Temperature 350K
Direction Specification Method Normal to Boundary
Turbulence Specification Method I11tensity and Hydraulic Diameter
Turbulence Intensity 3%
Hydraulic Diameter 92mm
The gauge pressure was measured experimentally, and entered relative to the operating pressure.
The turbulent intensity was kept constant, and the hydraulic diameter was the characteristic width at the outlet of the cascade.
6.3.2CONVERGENCE CONSJDERATJONS
To start the iteration process, all zones in the model were initialised to give the program a starting point with initial conditions that allow for the final solution to be attained after iteration.
FLUENT calculated average values for velocities, gauge pressure, turbulent kinetic energy and turbulence dissipation rates. The velocities were set manually to zero (in both the x- and y- directions) as it was found that this yielded faster convergence times in the simulations.
For the program to iterate successfully and achieve convergence, there must be some considerations which need to be taken into account. Once the initial guess is made, and an approximate solution is attained, this leads to a small imbalance in the conservation equation.
This imbalance in each cell in the model is referred to as the residual. When iterating, this value decays and begins to "level off' at a certain value. Judging convergence is usually done by monitoring the residuals, of which the values tend to drop in the order of 10-3.This is the default value set by FLUENT and was used in all the simulations. The program checks certain criteria with residual monitors and stops the simulation automatically once these criteria are met.
Together with these criteria, the drag coefficient and the mass flow balance at the inlet and outlet were monitored as well.
Another factor that affected the convergence of the simulation was the Courant number (CFL), which is the main control over the time-stepping scheme. The FLUENT user manual suggests a default CFL value of 5 for the coupled implicit solver, and possibly increasing it to la, 20, 100, or even higher. Lower CFLs are suggested at the start of the simulation, when changes in the solution are highly non-linear, and can be increased as the solution progresses as larger values lead to faster convergence. Different values for the CFL were found to be suitable for the different turbulence models, and are shown in Table 6-3.
Table 6-3: Suitable CFL Numbers for Different Turbulence Models
TURBULENCEMODEL CFL
Spalart-Allmaras 20
Realisable k- £ la
6.3.3MODELS AND MESHING
There were two geometric models used by De Villiers (2002). One included just the cascade box and blade geometries, while the other model incorporated the entire plenum space from the annular radiator and the cascade. The meshes used by De Villiers (2002) were refined such that the accuracy of results was increased and the convergence times improved. By observing the
"equiangle skewness" in GAMBIT, one could judge the improvement of the mesh by finding the cell with the worst skewness.
The first model that was adjusted was the grid involving only a 2-D model of the cascade box and SMR-95 blades, as shown in Figure 6-3. There are a number of meshing schemes that can be used, with several shapes of cells. The choice of scheme would depend on the complexity of the geometry, setup time and computational expense of the final grid.
Figure 6-3: Refined mesh using boundary layers
Instead of using a fully unstructured grid, a boundary layer mesh was applied to both the test blade and the neighbouring dummy blade. The resulting cell sizes on the walls of the suction and pressure sides of the test blade would result in more accurate predictions, and a better resolution. A boundary layer mesh was applied to the dummy blade as well, since the flow at the trailing edge affected the flow on the pressure side of the test blade. The previous grid of De Villiers (2002) had a first cell height of 0.01 mm, growth rate of 1.35 and 16 rows, which yielded a boundary layer height of 3.45 mm, and
l ::::::
1, which reduces stability problems in certain turbulence models. The worst cell skewness was 0.67.The boundary layer applied to the new mesh had a first cell height of 0.001 mm, growth factor of 1.2, and a larger number of rows of 27 as the cell sizes were chosen to be smaller. The pave meshing scheme was used to fill the cascade space for the main flow using quad elements of interval size 1.5. This yielded a worst equiangle skewness of 0.52, which was favourable compared to the previous cell skewness of De Villiers (2002).
To improve the resolution of the cells in the mesh, and possibly improve the accuracy of the results, a decomposed mesh was constructed. This involved splitting the domain into several sections such that the skewness would be minimal when using only quad cells. This mesh is shown in Figure 6-4.
Figure 6-4: Decomposed mesh
This mesh was achieved by using the same boundary layer mesh on the blades as used in the previous grid, and meshing the incomplete regions with quad cells using an interval size of 1.
This resulted in a total number of elements of close to 400 000, as compared to the total number of cells of around 18 500 for the previous grid. Cassie (2006) investigated the results of the simulations using this grid, and proved that the change in results was insignificant as compared to the original grid. The computational time was almost tripled, and thus the model was discarded, proving that the initial boundary layer mesh was sufficient for relatively accurate results.
for a smooth transition and minimal skew cells at the inlet of the cascade. The plenum was meshed using a very coarse mesh, as the flow in this area was not of close concern. The inlet conditions for this model would need to be known at the annular radiator, making this method unfavourable. It was shown by De Villiers (2002) that the difference in results of this plenum model proved to be insignificant compared to the original grid of the cascade box. Together with this knowledge and the fact that the inlet conditions were not accurately known, this model was discarded and the cascade box with definite inlet and outlet conditions was used.
Figure 6-5: Grid incorporating entire plenum
6.3.4RESULTS
The conditions and parameters were set in FLUENT and simulations were run for the free inlet condition using the different turbulence models. The initial blade wall temperatures were set at 313 K, as measured experimentally. By using the various CFL values for different turbulence models (as described in Section 6.3.2), FLUENT performed the iterations to attain a final solution that would calculate the heat flux into the blade along the pressure and suction surfaces, based on the flow conditions and the temperatures that would be calculated.
The conversion from the heat flux to a heat transfer coefficient was of the same method of the experimental calculation. This would involve dividing the wall heat flux by the difference between the total and wall temperatures at the point that would be calculated. The wall flux values were exported from FLUENT by coordinates, and using the x-axis for "curve length" and the y-axis for "wall fluxes".
The solutions for the various turbulence models all seemed to converge after around 2 000 iterations after using their specific CFL values. The Spalart-Allmaras model initially was not converging, and this non-convergence result was reported by De Villiers (2002). However, after lowering the CFL to around 5 in the initial stages of the simulation and gradually increasing this value back to 20 during the course of the iteration process, the solution converged. This method of maintaining a stable solution has been discussed previously in Section 6.3.2.
Heat Transfer Coefficient Distribution (Turb Int 3%)
--Tu=3% - S S T
SA RKE
-RNG -SKW
900 800
~N< 700
~E 600 IIc
..
'"
500E
..
~U... 400
~c
•
300!-...
;;
..
200::c
100 0
-110 -90 -70 -50 -30 -10 10 30
Circumferential distance (mm)
50 70 90 110 130 150
Figure 6-6: Heat transfer coefficient prediction for Tu= 3%
The heat transfer coefficient distribution along the SMR-95 blade that was predicted by each turbulence model is shown in Figure 6-6, together with the distribution that was calculated from experimental testing.
One can see that the general trend from thek-G models followed the experimental distribution with regard to the magnitude of the heat transfer coefficient on the pressure side. Although there is a slight under-prediction between 70 mm and 20 mm circumferential distance on the pressure side (and consequently a different curve), the heat transfer coefficient at the leading edge was accurately predicted. The Spalart-Allmaras model was the most accurate in predicting the leading edge heat transfer coefficient.
speculated point of another separation bubble. This phenomenon was unclear in the experimental heat transfer coefficient distribution due to reasons described in Section 5.3.5.4.
Further examination of the velocity vector plot at this point revealed a reversal in flow, as shown in Figure 6-7 and reported by De Villiers (2002) and Cassie (2006). Further turbulent regions and flow reversal were also noted from the vector plot at the trailing edge of the blade.
Figure 6-7: Flow reversal in the velocity vector plot indicating a separation bubble
The distribution on the suction side seemed to follow a similar trend compared to the experimental curve, however, there was a large over-prediction in the magnitude of the heat transfer coefficients for both the k-E models and k-(J) turbulence models. The slight disturbance in the curves of all the turbulence models at around 95 mm circumferential distance on the suction side was expected, as the experimental plot showed evidence of a smaller separation bubble at this point.
The SST k-(J) model seemed to follow the experimental distribution the most accurately on both the suction and pressure surfaces, as well as predicting the separation bubbles that occurred at various points along the SMR-95 surface. There was a large over-prediction in the magnitude of the heat transfer coefficient when compared to the experimental curve, and this proved to be somewhat constant throughout the distribution.
The standard k-(J) model predicted a distribution similar to that of the SST k-(J) turbulence model, with a few discrepancies when closely compared to the experimental distribution. The leading edge heat transfer coefficient was largely over-predicted as well, and proved to do so even further in the cases for Tu = 15% and Tu = 25.5%. Hence the model results for the standard k-(J) turbulence model will be omitted from those particular comparisons following.
Heat Transfer Coefficient Distribution (Turb Int 15%)
--Tu=15%
700
~ 600 N<
e 500
~:l c
..
400'ule
..
Clu
..
300~c
..
~ 200
... ..
;::
100
0
·110 -90 -70 -50 ·30 ·10 10 30 50
- S S T - S A -RNG
70 90
RKE
110 130 150
Circumrerenlial distance (mm)
Figure 6-8: Heat transfer coefficient prediction for Tu== 15%
Heat Transfer Coefficient Distribution (Turb Int 25.5%)
• Tu=25.5% --55T
SA RKE
-RNG
150 130 110 90
.. .
...
70 50
'
... .. .... .
30 10 -10 -30 -50 800
:;2' 700
N<
E 600
~
.,C
..
500 l:U 400... ..
ClU
..
.;
300..
c Eo-..
200... ..
:c 100
0
-110 -90 ·70
Circumferential distance (mm)
Figure 6-9: Heat transfer coefficient prediction for Tu==25.5%
The heat transfer coefficient distributions shown in Figure 6-8 and Figure 6-9 depict the
The k-& models still represent a closer prediction to the experimental curve with regard to the magnitude of heat transfer. The actual curves do not match the experimental distributions for both the Tu
=
15% and Tu=
25.5% cases, and again there is a large over-prediction on the suction side of the blade. The Spalart-Allmaras model again accurately predicted the heat transfer coefficient at the leading edge of the blade for both turbulence conditions when compared to the free inlet condition. One can see in the results for Tu = 15% that the disturbances (occurring at the discussed points where separation bubbles occur) have diminished slightly. This prediction is representative of the actual occurrence in the experimental plot as discussed in Section 5.3.5.5. The SST k-w turbulence model again predicts a more accurate distribution, but the magnitude of heat transfer is shown to be largely over- predicted, especially at the leading edge of the blade.The discontinuity of the distribution curve on the suction side of the SST k-w model at approximately 90 mm circumferential distance gives a good indication of the diminishing phenomenon of the drop in heat transfer at this point. The lack of continuity of the experimental distribution along this range gives an indication of the almost crude nature of the data acquisition system, and the various experimental errors that could have affected the results.
By comparing this directly with the case for Tu = 25.5%, one can see the lack of drop in heat transfer at the points near the leading edge where the separation bubbles occur for lower turbulence intensities. The indication of a small bubble near the trailing edge of the blade is now almost completely absent when looking at the SST k-w turbulence model.
6.3.5CONCLUSIONS
From the different geometries of incorporating the entire plenum and decomposing the mesh for the cascade geometry, only one model was deemed accurate and reliable enough to attain heat transfer coefficient distribution predictions along the SMR-95 turbine blade to compare to the experimental data set.
The cascade model was used and the boundary layer mesh was improved together with the pave meshing scheme. With the inlet and outlet conditions accurately known, these were applied to the model with the relevant CFL values that seemed to optimise each turbulence model. The resulting solutions from FLUENT provided insight into the different predictions of the various turbulence models.
The trend seemed consistent for all the turbulence intensity cases, with the k-E turbulence models proving to be the most accurate in predicting the heat transfer close to the leading edge of the blade. The Spalart-A11maras model seemed to accurately predict the heat transfer coefficient at the leading edge. Both the Realisablek-E model and the RNG k-E model lacked the ability to closely predict the distribution curve on both the pressure and suction sides of the blade, when compared to the experimental distribution. There was a large over-prediction of the magnitude of heat transfer on the suction side for all turbulence models.
The SST k-w turbulence model proved to be the most accurate in predicting the actual curve compared to the experimental distribution, however, it over-predicted the magnitude of heat transfer along the entire surface of the blade and even more at the leading edge of the blade.
This shows the model's strengthinpredicting flows in transonic conditions, but did not perform due to the conditions of large flow separation. This is a commonly-known disadvantage of this turbulence model, as it uses the Boussinesq approximation which results in an inability to predict anisotropy (Pattijnet al. (1999)).
The discrepancies in the CFD results are believed to be the result of using a 2-D model where the secondary flow effects are not taken into account, and of course the various experimental errors that could have contributed to the inaccuracies of the experimental heat transfer distribution. Cassie (2006) used a 3-D geometry to ascertain whether there would be a difference in the prediction of heat transfer for the turbulence models. He showed that there was no significant difference on the pressure surface of the blade. The presence of horseshoe vortices, which was also reported by Graziani et al. (1980), was shown to increase the heat transfer slightly on the suction side. Since the experimental data is taken at the midspan of the blade, there is not enough of a significant difference in the heat transfer prediction between the 2-D and 3-D models to validate the extremely large computational times needed for the 3-D simulations.
The static pressure contour plots for all the turbulence models are gIven In Appendix B, showing the visual differences in the predictions for the models with the various turbulence intensities.
CHAPTER
7
CONCLUSIONS